One of the more challenging indefinite integrals of elementary calculus is In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...
Derivation
This antiderivative may be found by integration by parts, as follows: In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...
where
Then
Next we add to both sides of the equality just derived:
Then divide both sides by 2.
Higher odd powers of secant
Just as the integration by parts above reduced the integral of secant cubed to the integral of secant to the first power, so a similar process reduces the integral of higher odd powers of secant to lower ones.
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